Syllabus Honors Geometry
AP Calculus AB Syllabus
Instructor: Paula Ortiz
Phone: 281- 641 6661
Email: paula.ortiz@humble.k12.tx.us
Course Description:
Calculus assimilates many of the concepts of previous mathematics courses, and demonstrates the relevance of the material taught prior to calculus. Calculus AB provides a strong foundation for university-level mathematics and science courses. The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.
First Semester:
Chapter 1: Limits and Their Properties (9 days – one test)
1. • An introduction to limits, including an intuitive understanding of the limit process
2. • Using graphs and tables of data to determine limits
3. • Properties of limits
4. • Algebraic techniques for evaluating limits
5. • Comparing relative magnitudes of functions and their rates of change
6. • Continuity and one-sided limits
7. • Geometric understanding of the graphs of continuous functions
8. • Intermediate Value Theorem
9. • Infinite limits
10. • Using limits to find the asymptotes of a function; discussion of end-behavior of functions
Chapter 2: Differentiation (14 days – three tests)
1. • Zooming-in activity and local linearity
2. • Understanding of the derivative: graphically, numerically, and analytically
3. • Approximating rates of change from graphs and tables of data
4. • The derivative as: the limit of the average rate of change, an instantaneous rate of
5. change, limit of the difference quotient, and the slope of a curve at a point
6. • The meaning of the derivative - translating verbal descriptions into equations
7. • The slope of a curve at a point and the slope of the tangent line to a curve at a point.
1. • The relationship between differentiability and continuity
2. • Functions that have a vertical tangent at a point
3. • Functions that have a point on which there is no tangent
4. • Differentiation rules for basic functions, including power functions and trig functions
5. • Rules of differentiation for sums, differences, products, and quotients
6. • The chain rule
7. • Implicit differentiation
8. • Related rates
9.
Chapter 5: Derivatives of Logarithmic and Exponential Functions (8 days – one test)
• The natural logarithmic function and differentiation
• Exponential functions: differentiation
Chapter 3: Applications of Differentiation (25 days – five tests)
1. • Extrema on an interval and the Extreme Value Theorem
2. • Rolle’s Theorem and the Mean Value Theorem, and their geometric consequences
3. • Increasing and decreasing functions and the First Derivative Test
4. • Concavity and its relationship to the first and second derivatives
5. • Second Derivative Test
6. • Limits at infinity
7. • A summary of curve sketching—using geometric and analytic information as well as
8. calculus to predict the behavior of a function
9. • Relationship between the increasing and decreasing behavior of [pic]and the sign of [pic].
10. • Relationship between the concavity of [pic] and the sign of[pic].
11. • Relating the graphs of [pic]
12. • Optimization including both relative and absolute extrema
13. • Tangent line to a curve and linear approximations
14. • Application problems including position, velocity, acceleration, and rectilinear motion
First Semester Exam (two review days)
Second Semester:
Chapter 4: Integration (18 days—three tests)
1. • Antiderivatives and indefinite integration, including antiderivatives following directly from
2. derivatives of basic functions
3. • Basic properties of the definite integral
4. • Area under a curve
5. • Meaning of the definite integral
6. • Definite integral as a limit of Riemann sums
7. • Riemann sums, including left, right, and midpoint sums
8. • Trapezoidal sums
1. • Use of Riemann sums and trapezoidal sums to approximate definite integrals of
2. functions that are represented analytically, graphically, and by tables of data
3. • Discovery lesson on the First Fundamental Theorem of Calculus
4. • Use of the First Fundamental Theorem to evaluate definite integrals
5. • Use of substitution of variables to evaluate definite integrals
6. • Integration by substitution
7. • Discovery lesson on the Second Fundamental Theorem of Calculus
8. • The Second Fundamental Theorem of Calculus and functions defined by integrals
9. • The Mean Value Theorem for Integrals and the average value of a function
Chapter 5: Logarithmic, Exponential, & Other Transcendental Functions (19 days – four tests)
1. • The natural logarithmic function and integration
2. • Inverse functions
3. • Exponential functions: integration
4. • Bases other than e and applications
5. • Solving separable differential equations
6. • Applications of differential equations in modeling, including exponential growth
7. • Use of slope fields to interpret a differential equation geometrically
8. • Drawing slope fields and solution curves for differential equations
1. • Inverse trig functions and differentiation
2. • Inverse trig functions and integration
Chapter 6: Applications of Integration (8 days – two tests)
1. • The integral as an accumulator of rates of change
2. • Area of a region between two curves
3. • Volume of a solid with known cross sections
4. • Volume of solids of revolution
5. • Applications of integration in problems involving a particle moving along a line,
6. including the use of the definite integral with an initial condition and using the definite
7. integral to find the distance traveled by a particle along a line
Chapter 7: Integration Techniques (4 days – one test)
1. • Review of basic integration rules
2. • Integration by parts
3.
Review for the AP Exam
(3 weeks – three tests)
After the AP Exam:
(9 days – two tests)
Additional topics that are not listed as required topics in the Calculus AB Topic Outline in the AP Calculus Course Description are addressed. Such topics are:
4. • Finding volume using the shell method
5. • L’Hopital’s rule.
6. • Integration by partial fractions
• Work done by a constant force and work done by a variable force.
Projects such as the following are assigned to emphasize and reinforce the calculus used for such:
-Larson’s ideas for drawing characters or objects on the calculator using
calculus statements. Many kinds of stipulations and restrictions can
be put on the number and kind of statements.
-Construct models that illustrate how to find volume using cross-sections.
-Writing calculus programs using Winplot for computation and drawings.
Second Semester Exam (two review days)
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